Linear Functions – Concretely, Pictorially, Symbolically

Teachers often talk about how manipulatives can help the struggling learner. I’m suggesting that having students solve problems concretely can assist all learners.

When I experienced this problem using the blue and yellow tiles, I gained a deeper understanding of the problem. The equation y = 3x + 2 now had meaning. I was able to find the pattern in the table to determine the number 3. By modelling the problem using tiles, I was able to see this as adding an extra 3 blue tiles every time the figure grew.

In the past, I had difficulty explaining to students where the 2 came from. I could convince them that it had to be there. For example, take the point (2, 8). Multiplying the 2 by 3 gives 6, so we need to add 2 more. Looking at this concretely & pictorially, the 2 now has meaning. For me, it is how many blue tiles there were before we start adding yellow & blue tiles. (See the photo below.)

Your students who used to get it symbolically will still get it if they approach it concretely. However, what it means to “get it” in your classroom will start to change.

Patterning the Blues Concretely

I’d appreciate your comments. Maybe you have some thoughts on how this activity addresses the 7 processes?

When I did this activity with my students it really made connections for them! Students were most impressed with being able to “SEE” where those numbers on the y=3x+2 were coming from! Often times they say the pattern is “add 3″ but this method provided for them a visual check and reminder of the EXTRA 2!!!
For my younger grades I didn’t introduce the graph portion but my grade 10’s again appreciated being able to put those lines into CONTEXT.